Malay Banerjee

Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system

Mechanisms and scenarios of pattern formation in predator–prey systems have been a focus of many studies recently as they are thought to mimic the processes of ecological patterning in real-world ecosystems. Considerable work has been done with regards to both Turing and non-Turing patterns where the latter often appears to be chaotic. In particular, spatiotemporal chaos remains a controversial issue as it can have important implications for population dynamics. Most of the results, however, were obtained in terms of ‘traditional’ predator–prey models where the per capita predation rate depends on the prey density only. A relatively new family of ratio-dependent predator–prey models remains less studied and still poorly understood, especially when space is taken into account explicitly, in spite of their apparent ecological relevance. In this paper, we consider spatiotemporal pattern formation in a ratio-dependent predator–prey system. We show that the system can develop patterns both inside and outside of the Turing parameter domain. Contrary to widespread opinion, we show that the interaction between two different type of instability, such as the Turing–Hopf bifurcation, does not necessarily lead to the onset of chaos; on the contrary, the emerging patterns remain stationary and almost regular. Spatiotemporal chaos can only be observed for parameters well inside the Turing–Hopf domain. We then investigate the relative importance of these two instability types on the onset of chaos and show that, in a ratio-dependent predator–prey system, the Hopf bifurcation is indeed essential for the onset of chaos whilst the Turing instability is not.

Bifurcation analysis of a ratio-dependent prey–predator model with the Allee effect

A B S T R A C T There is a growing body of evidence supporting implementation of ratio-dependent functional response of predators in ecological models. Those models often provide a satisfactory explanation of the observed patterns of dynamics which cannot be done based on the ‘classical’ models using the prey-dependent functional response. Surprisingly enough, all theoretical analysis of ratio-dependant predator–prey interactions has so far been completed only for the simplest case where the prey growth is logistic. In a large number of ecologically relevant situations, however, the growth rate of a population is subject to an Allee effect and the per capita growth rate increases with population density. Taking into account Allee dynamics for the prey growth in models can alter the previous theoretical findings obtained for the logistic growth paradigm. In this paper, we analyse a ratio-dependent predator–prey system with prey growth subject to an Allee effect. We both consider the cases of a strong Allee effect (the population growth rate is negative at low species density) and the case of a weak Allee effect (the population growth is positive at low population density). For both cases we fulfil a comprehensive bifurcation analysis, constructing the parametric diagrams, and also show possible phase portraits. Then we compare the properties of the ratio-dependant predator–prey model with and without the Allee effect and show a substantial difference in the dynamical behaviour of those systems. We show that including an Allee effect in a ratio-dependent predator–prey model removes the possibility of sustainable oscillations of species densities (population cycles). We show that the ratio-dependent predator–prey model with the Allee effect can solve the paradox of enrichment. However, unlike the same model with logistic growth, incorporating the Allee effect results in a paradox of biological control.

Turing instabilities and spatio-temporal chaos in ratio-dependent Holling–Tanner model

In this paper we consider a modified spatiotemporal ecological system originating from the temporal Holling-Tanner model, by incorporating diffusion terms. The original ODE system is studied for the stability of coexisting homogeneous steady-states. The modified PDE system is investigated in detail with both numerical and analytical approaches. Both the Turing and non-Turing patterns are examined for some fixed parametric values and some interesting results have been obtained for the prey and predator populations. Numerical simulation shows that either prey or predator population do not converge to any stationary state at any future time when parameter values are taken in the Turing-Hopf domain. Prey and predator populations exhibit spatiotemporal chaos resulting from temporal oscillation of both the population and spatial instability. With help of numerical simulations we have shown that Turing-Hopf bifurcation leads to onset of spatio-temporal chaos when predators diffusivity is much higher compared to prey population. Our investigation reveals the fact that Hopf-bifurcation is essential for the onset of spatiotemporal chaos.

Dynamical analysis of fractional-order modified logistic model

In this paper, we study a fractional differential equation model of the single species multiplicative Allee effect. First we study the stability of equilibrium points. Further we give some sufficient conditions ensuring the existence and uniqueness of integral solution. In the last section we perform several numerical simulations to validate our analytical findings.

Self-replication of spatial patterns in a ratio-dependent predator-prey model

The results concerning the self-replication pattern formation in the spatio-temporal prey-predator model with ratio-dependent functional response are reported. The Turing instability region is obtained with the help of standard analysis of the linearized model around the coexisting equilibrium point. Numerical simulation reveals the self-replicating pattern for a certain choice of parametric values.

Dynamics of additional food provided predator-prey system with mutually interfering predators.

Use of additional/alternative food source to predators is one of the widely recognised practices in the field of biological control. Both theoretical and experimental works point out that quality and quantity of additional food play a vital role in the controllability of the pest. Theoretical studies carried out previously in this direction indicate that incorporating mutual interference between predators can stabilise the system. Experimental evidence also point out that mutual interference between predators can affect the outcome of the biological control programs. In this article dynamics of additional food provided predator-prey system in the presence of mutual interference between predators has been studied. The mutual interference between predators is modelled using Beddington-DeAngelis type functional response. The system analysis highlights the role of mutual interference on the success of biological control programs when predators are provided with additional food. The model results indicate the possibility of stable coexistence of predators with low prey population levels. This is in contrast to classical predator-prey models wherein this stable co-existence at low prey population levels is not possible. This study classifies the characteristics of biological control agents and additional food (of suitable quality and quantity), permitting the eco-managers to enhance the success rate of biological control programs.

Spatial pattern formation in ratio-dependent model: higher-order stability analysis

The article presents a study of the spatiotemporal pattern formation in a Holling-Tanner prey-predator model with ratio-dependent functional response. Conditions for Turing bifurcation are obtained and different spatially inhomogeneous stationary patterns exhibited by the model system are presented. Then, reported patterns are the outcome of numerical simulation. The conditions for instability of inhomogeneous spatiotemporal perturbation around temporal steady state beyond the linear regime fall out from the analysis of higher-order perturbation terms. The analytical findings are validated with the numerical simulation results.

A comparative study of deterministic and stochastic dynamics for a non-autonomous allelopathic phytoplankton model

Abstract In this paper, we investigate a non-autonomous competitive phytoplankton model with periodic coefficients in deterministic and stochastic environment, respectively. We prove the existence of at least one positive periodic solution together with it’s global asymptotic stability. The existence of periodic solution has been obtained by using the continuation theorem of coincidence degree theory proposed by Gaines and Mawhin. We formulate the corresponding stochastic model by perturbing the growth rate parameters by white noise terms. We prove that all the higher order moments of the solution to the stochastic system is uniformly bounded which ensure that the solution of the stochastic system is stochastically bounded. We provide easily verifiable sufficient conditions for non-persistence in mean, extinction and stochastic permanence of the stochastic system. Sufficient condition for permanence shows that if the noise intensity is very low then the solution of the stochastic system persists in the periodic coexistence domain of the deterministic system. We perform exhaustive numerical simulations to validate our analytical findings.

Prey-predator model with a nonlocal consumption of prey

The prey-predator model with nonlocal consumption of prey introduced in this work extends previous studies of local reaction-diffusion models. Linear stability analysis of the homogeneous in space stationary solution and numerical simulations of nonhomogeneous solutions allow us to analyze bifurcations and dynamics of stationary solutions and of travelling waves. These solutions present some new properties in comparison with the local models. They correspond to different feeding strategies of predators observed in ecology.

Global dynamics of an additional food provided predator–prey system with constant harvest in predators

Abstract The article aims to study the global dynamics associated with a predator prey system when the predator is provided with additional food and harvested at a constant rate. This study supplements the existing literature on the dynamics of additional food provided predator prey system by focusing on the consequences of harvesting the predators. It presents a comprehensive view on the entire range of bifurcations that take place in the considered system and highlights the dependence of the system dynamics on its vital parameters. This study provides important tools for investigations pertaining to controllability of the system which are essential from the real world applications perspective.